The discussions are aimed at improving bilateral relations that have been strained by nearly 14 years of war and America`s often testy relations with former Afghan President Hamid Karzai.
On Tuesday, before meeting Obama, Ghani will head to Arlington National Cemetery, where many of the 2,300 US troops killed in Afghanistan are buried.
Pace of withdrawal
Roughly 9,800 American troops remain stationed in Afghanistan to advise and train Afghan troops after the last U.S. combat forces withdrew late last year.
Obama has announced plans to cut the troop level to 5,500 by year`s end and even less by the time he leaves office in early 2017. The pace of withdrawal is a key element of the two countries` discussions.
During the Monday talks, the leaders addressed Ghani`s concerns regarding security, ahead of the spring months that have traditionally seen more frequent than usual Taliban attacks.
Obama is "actively considering" Ghani`s request that there be some flexibility in the withdrawal of U.S. troops, Kerry told reporters Monday, after a day of talks in the presidential woodland retreat of Camp David in Maryland.
To help shore up stability, the United States announced it would ask Congress for funding to allow the Afghan National Security Force to maintain the 352,000 troop level through at least fiscal year 2017.
Both sides also agreed to require the Afghan government to complete specific reforms and meet other milestones in order to receive up to $800 million in economic aid. U.S. officials said the Afghans suggested the incentive-based funding idea.
Ghani, asked by VOA about what he sees as adequate U.S. troop strength, declined to give a number.
"The question on numbers is a decision for the president of the United States, and that decision will solely be made by President Obama," Ghani said.
"What we have emphasized and agreed is that we are strategic partners," he said at the briefing. Deferring to "the judgments of experts" including Pentagon personnel, he added, "Numbers are a means, they are not an ending of themselves."
Neither Kerry nor Defense Secretary Ashton Carter would comment on a slowed timeline for the U.S. drawdown, saying this would be discussed during the Tuesday talks.
Also at stake: the future of U.S. bases in Jalalabad and in Kandahar, where the Taliban had their capital until 2001. U.S. military leaders have seemed receptive to Ghani`s request that those bases stay open as long as possible, The Associated Press reported Tuesday.
Earlier Monday, Ghani said that Afghanistan`s "partnership with the United States is foundational" and thanked the U.S. for giving his country "freedom and hope."
"We are delighted to have the opportunity to discuss a changing context and to be able to affirm a partnership that is based on values, respect for democratic process, electoral reforms, comprehensive reforms of the economy, governance, and related issues," Ghani said.
During a stop at the Pentagon Monday, Ghani warmly thanked U.S. troops for more than a decade of sacrifice since the 2001 toppling of a Taliban government by a U.S.-led invasion.
The new Afghan leader vowed to defeat Taliban insurgents seeking to overthrow his fledgling government in fighting that continues more than 13 years after the United States invaded Afghanistan to attack al-Qaida militants after their 2001 terrorist assault on the U.S.
"Terrorism is a threat, it`s evil," Ghani said as he greeted Defense Secretary Carter at the Pentagon. "But we the people of Afghanistan are willing to speak truth to terror by saying, `No. You will never overwhelm us. You will never subdue us. We are going to overcome.` And in this endeavor our partnership with the United States is foundational because we will be the first line of defense for freedom globally."
In a stark reminder of the violence still racking Afghanistan, gunmen killed 13 bus passengers in a province near Kabul early Tuesday.
Ghani also is expected to address a joint meeting of Congress Wednesday. On Thursday, the Afghan leaders head to New York for meetings with United Nations officials.